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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.2.19a
Chapter 3, Problem 3.2.19a

Use the graph of f in the figure to do the following. <IMAGE>
a. Find the values of x in (-2,2) at which f is not continuous.

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Step 1: Understand the concept of continuity. A function f(x) is continuous at a point x = a if the following three conditions are met: (1) f(a) is defined, (2) the limit of f(x) as x approaches a exists, and (3) the limit of f(x) as x approaches a is equal to f(a).
Step 2: Examine the graph of f(x) over the interval (-2, 2). Look for any points where the function might not be continuous. These could be points where there is a jump, a hole, or a vertical asymptote in the graph.
Step 3: Identify any points of discontinuity. A point of discontinuity is where at least one of the conditions for continuity is not satisfied. Check for any breaks in the graph, such as gaps or jumps.
Step 4: Verify the type of discontinuity at each identified point. Determine if the discontinuity is removable (a hole in the graph), a jump (a sudden change in function value), or an infinite discontinuity (a vertical asymptote).
Step 5: List the x-values in the interval (-2, 2) where the function f(x) is not continuous, based on your observations from the graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. Discontinuities can occur due to jumps, holes, or vertical asymptotes in the graph.
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Types of Discontinuities

There are several types of discontinuities: removable (or hole), jump, and infinite. A removable discontinuity occurs when a function is not defined at a point but can be made continuous by defining it appropriately. A jump discontinuity occurs when the left-hand and right-hand limits exist but are not equal, while an infinite discontinuity occurs when the function approaches infinity at a point.
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Analyzing Graphs for Continuity

To determine where a function is not continuous using its graph, one must look for breaks, jumps, or asymptotes. Points where the graph does not connect smoothly indicate discontinuities. By examining the behavior of the graph as it approaches specific x-values, one can identify intervals where the function fails to be continuous.
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Related Practice
Textbook Question

Vertical tangent lines

a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).

Textbook Question

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by y=10e−t2cos(πt8)y=10e^{-\(\frac{t}{2}\)}\(\cos\[\left\)(\(\frac{\pi t}{8}\]\right\)).

a. Graph the displacement function. 

Textbook Question

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

a. Graph the volume function. What is the volume of water in the tank before the valve is opened? 

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Textbook Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = -7x; P(-1,7)

Textbook Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 1/x; P (1,1)

Textbook Question

Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.

73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.

a. f(x) = (x-2)^1/3